Irisan Kerucut - Parabola • Part 4: Persamaan Parabola yang Berpuncak di Titik P(a,b)

Introduction

This tutorial focuses on understanding the equation of a parabola with its vertex at point P(a, b). It serves as a continuation of a series on conic sections, specifically parabolas, and aims to provide detailed derivations and practical examples to enhance your understanding of this mathematical concept.

Step 1: Understand the Vertex Form of the Parabola

The vertex form of a parabola is essential for identifying its characteristics. The general equation for a parabola with its vertex at point P(a, b) can be expressed as:

• For a vertical parabola: [ y = k(x - a)^2 + b ]

• For a horizontal parabola: [ x = h(y - b)^2 + a ]

Key points

• Vertex (a, b): This is the highest or lowest point of the parabola, depending on its orientation.
• k: This coefficient determines the direction and width of the parabola. If k > 0, the parabola opens upward (for vertical) or to the right (for horizontal). If k < 0, it opens downward (for vertical) or to the left (for horizontal).

Step 2: Identifying the Orientation of the Parabola

Determining whether the parabola opens vertically or horizontally is crucial for using the correct formula.

• Vertical Parabola: Use the first equation (y = k(x - a)^2 + b).
• Horizontal Parabola: Use the second equation (x = h(y - b)^2 + a).

Practical Tips

• To decide on the orientation, look at the context of the problem or graph if available.
• For real-world applications, vertical parabolas can represent projectile motions, while horizontal parabolas can model satellite dishes.

Step 3: Example Derivation

Let’s derive an example for a vertical parabola with vertex P(2, 3).

1. Start with the vertex form: [ y = k(x - 2)^2 + 3 ]

2. Determine k based on additional information, such as another point the parabola passes through. If we know it passes through (4, 7), we can substitute: [ 7 = k(4 - 2)^2 + 3 ] [ 7 = 4k + 3 ] [ 4 = 4k \implies k = 1 ]

3. Final equation: [ y = (x - 2)^2 + 3 ]

Step 4: Graphing the Parabola

Once you have the equation, graphing it provides visual insights.

• Plot the vertex: Start at point P(2, 3).
• Identify the direction: Since k = 1, the parabola opens upward.
• Draw the axis of symmetry: This is a vertical line through the vertex, (x = 2).
• Choose additional points: For example, calculate y-values for x = 1 and x = 3 to find points on the parabola.

Common Pitfalls

• Ensure you correctly identify k's sign; this will affect the parabola's orientation.
• Double-check calculations when deriving k from known points.

Conclusion

In this tutorial, you learned about the equation of a parabola with its vertex at point P(a, b), how to derive it, and how to graph it effectively. The key takeaway is understanding the vertex form and the significance of the vertex in determining the parabola's characteristics. For further study, consider exploring more examples and different types of conic sections related to parabolas.